Mathematics# Associative property

## What is associative property?

## History

## Associative property of the addition

## Associative property of multiplication

## Examples

**Associative property** can be found in **Algebra** and can be applied to two different types of operations: **addition** and **multiplication**. This mathematical property certainly indicates that, when there are three or more numbers in these operations, the **result** does not depend on how the **digits** are **placed** or **grouped**. This means that, no matter how the different numbers come together in a given operation, addition or multiplication will always give the same **result** no matter the **order**. Grouping, in this way, has no relation to the result obtained in the mathematical operation.

**Related topics**

Associative property is a **property** within the area of **algebra** that can be applied to **additions** and **multiplications**. It tells us that when there are three or more numbers in an operation, the **result** will not depend on how the terms are **placed**.

In **1830**, the **Algebra Treaty** was published which tried to explain the term as a logical treatment comparable to **Euclid’s** elements. He spoke of two different types of **algebra, arithmetic algebra and symbolic algebra**. In the book, he describes symbolic algebra as the science that treats **combinations** of arbitrary **signs** and symbols by defined means through arbitrary laws. The truth is that it is very difficult to give an exact date on which it was created because people already knew that, for example, 2 + 3 = 3 + 2 from antiquity, but eventually people realized that this was a **general** **property** that could be attributed to other operations apart from addition and multiplication, and then it became something object of study. It can be said that it was not just one person who made this discovery.

The associative property of **addition** or **sum** establishes that the change in the **order** in which the numbers are **added** does not affect the **result** of the addition. Since the application of the associative property in addition has no apparent or important effect on itself, some doubts may arise about its usefulness and importance, however, having knowledge about these principles is useful for us to **perfectly master** these **operations**, especially when combined with others, such as subtraction and division; and even more so in the case of division to achieve the correct use of **mathematics**.

Multiplication is a mathematical operation that has different types of properties. One of them is the property that indicates that the way of grouping the factors will not cause any type of alteration in the final result of the multiplication regardless of the number of factors found in the operation.

As first example we are going to carry out the operation: **5 x 4 x 2**

The first thing we must do is to **group** the first **two** numbers, in this case they will be 5 and 4. Performing this step, then we will obtain the following equation:

(5 x 4) x 2

20 x 2

40

Now, if we **group** 4 and 2, we get the next result:

5 x (4 x 2)

5 x 8

40

As can be clearly seen in the previous operation, although the numbers were positioned differently, the **result remained the same**. Another example we can cite is the following:

( 2 x 3 ) x 5 = 2 x ( 3 x 5 )

6 x 5 = 2 x 15

30 = 30

Written by Gabriela Briceño V.

Briceño V., Gabriela. (2019). *Associative property*. Recovered on 5 October, 2023, de Euston96: https://www.euston96.com/en/associative-property/